Exam 13: Vector Functions

Find the limit $\lim _ { t \rightarrow \infty } \left\langle e ^ { - 4 t } , \frac { 1 } { t } , \frac { 7 t ^ { 2 } } { t ^ { 2 } + 1 } \right\rangle$ .

Find the unit tangent vector $T ( t )$ . $\mathbf { r } ( t ) = \langle 2 \sin t , 4 t , 2 \cos t \rangle$

The curvature of the curve given by the vector function $r$ is $k ( t ) = \frac { \left| \mathbf { r } ^ { \prime } ( t ) \times \mathbf { r } ^ { \prime \prime } ( t ) \right| } { \left| \mathbf { r } ^ { \prime } ( t ) \right| ^ { 3 } }$ Use the formula to find the curvature of $\mathbf { r } ( t ) = \left\langle \sqrt { 13 } t , e ^ { t } , e ^ { - t } \right\rangle$ at the point $( 0,1,1 )$ .

Find the scalar tangential and normal components of acceleration of a particle with position vector $\mathbf { r } ( t ) = 3 \sin t \mathbf { i } + 3 \cos t \mathbf { j } + 5 t \mathbf { k }$

Find the domain of the vector function $\mathbf { r } ( t ) = 8 t \mathbf { i } + \frac { 1 } { t - 4 } \mathbf { j }$ .

Find the length of the curve $\mathbf { r } ( t ) = - 2 t \mathbf { i } - t \mathbf { j } + t \mathbf { k }$ $- 2 \leq t \leq 1 .$

Find $\mathbf { r } ^ { \prime } ( t )$ and $\mathbf { r } ^ { \prime \prime } ( t )$ for $\mathbf { r } ( t ) = 5 t \mathbf { i } + 4 t ^ { 2 } \mathbf { j } + 5 t ^ { 3 } \mathbf { k }$

A particle moves with position function $\mathbf { r } ( t ) = 4 \sqrt { 2 } t \mathbf { i } + e ^ { 4 t } \mathbf { j } + e ^ { - 4 t } \mathbf { k }$ . Find the acceleration of the particle.

Find the acceleration of a particle with the given position function. $\mathbf { r } ( t ) = 9 \sin t \mathbf { i } + 10 t \mathbf { j } - 8 \cos t \mathbf { k }$

Sketch the curve of the vector function $\mathbf { r } ( t ) = \left\langle t ^ { 2 } , t ^ { 3 } , t \right\rangle$ $t \geq 0$ and indicate the orientation of the curve.

Let $\mathbf { r } ( t ) = \left\langle \sqrt { 6 - t } , \frac { \left( e ^ { t } - 2 \right) } { t } , \ln ( t + 1 ) \right\rangle$ . Find the domain of $r$ .

Find $\mathbf { r } ^ { \prime } ( t )$ and $\mathbf { r } ^ { \prime \prime } ( t )$ for $\mathbf { r } ( t ) = \langle t \cos 3 t - \sin 3 t , t \sin 4 t + \cos 4 t \rangle$

Two points A and B are located 100 ft apart on a straight line. A particle moves from A toward B with an initial velocity of 9 ft/sec and an acceleration of $\frac { 1 } { 2 }$ $\mathrm { ft } / \mathrm { sec } ^ { 2 }$ . Simultaneously, a particle moves from B toward A with an initial velocity of 2 ft/sec and an acceleration of $\frac { 3 } { 4 }$ $\mathrm { ft } / \mathrm { sec } ^ { 2 }$ . When will the two particles collide? At what distance from A will the collision take place?

Find $\mathbf { r } ( t )$ satisfying the conditions for $\mathbf { r } ^ { t } ( t ) = 9 e ^ { 9 t } \mathbf { i } + 9 e ^ { - t } \mathbf { j } + e ^ { t } \mathbf { k }$ $\mathbf { r } ( 0 ) = \mathbf { i } - \mathbf { j } + 9 \mathbf { k }$

For the curve given by $\mathbf { r } ( t ) = (4 \sin t , 5 t , 4 \mathrm { cos } t )$ , find the unit normal vector. $\mathbf { a } ( t ) = 2 \mathbf { k } , \mathbf { v } ( 0 ) = 10 \mathbf { i } - 9 \mathbf { j }$

Find the point(s) on the graph of $y = e ^ { - 11 x ^ { 2 } }$ at which the curvature is zero.

The helix $\mathbf { r } _ { 1 } ( t ) = 8 \cos t \mathbf { i } + \sin t \mathbf { j } + t \mathbf { k }$ intersects the curve $\mathbf { r } _ { 2 } ( t ) = ( 8 + t ) \mathbf { i } + 10 t ^ { 2 } \mathbf { j } + 9 t ^ { 3 } \mathbf { k }$ at the point $( 8,0,0 )$ . Find the angle of intersection.

Find a vector function that represents the curve of intersection of the two surfaces: The circular cylinder $x ^ { 2 } + y ^ { 2 } = 9$ and the parabolic cylinder $z = x y$ .

Find the point of intersection of the tangent lines to the curve $\mathbf { r } ( t ) = ( \sin \pi t , 5 \sin \pi t , \cos \pi t )$ , at the points where $t = 0$ and $t = 0.5$ .

Find the curvature of the curve $\mathbf { r } ( t ) = 2 t \mathbf { i } + 6 t \mathbf { j } + 9 \mathbf { k }$ .